AREA OF EQUILATERAL TRIANGLE

Area of equilateral triangle can be found using the formula given below.

Area of Equilateral triangle  =  (√3/4) a²

Here a stands for the length of side of a triangle.

What is equilateral triangle ?

In geometry, an equilateral triangle is a triangle in which all three sides are equal. The area enclosed by this shape is known as area of equilateral triangle.

Deriving the formula to find the area of equilateral triangle :

Take an equilateral triangle of the side “a” units. Then draw a perpendicular bisector to the base of height “h”.

By drawing perpendicular from A, we get two congurent right triangle ABD and ADC.

Area of triangle ABC  =  Area of triangle ABD + Area of triangle ADC

Since triangles ABD and ADC are congurent, areas will be equal.

Area of triangle ABC  =  Area of ABD + Area of ADC

Area of triangle ABC  =  2  (Area of ABD)

=  2 ⋅ (1/2) ⋅ Base ⋅ Height

Area of triangle ABC  =  Base ⋅ Height  ---(1)

In triangle ABD,

Base (BD)  =  a/2 and height (AD)  =  h

Using Pythagorean theorem,

a²  =  h² + (a/2)²

h²  =  a² - (a²/4)

h²  =  (3a²/4)

h  =  √(3a²/4)

h  =  (a√3/2)

By applying the values of base and height in (1), we get

Area of triangle ABC  =  (a/2) ⋅ (a√3/2)

  =  (1/4)a²√3

Area of triangle ABC  =  √3a²/4

Let us see some examples problems to understand how to find the area of the equilateral triangle.

Example 1 :

Find the area of the equilateral triangle having the length of the side equals 10 cm.

Solution :

Area of equilateral triangle  =  (√3/4) a²

Here a = 10 cm

  =  (√3/4) (10)²

  =  (√3/4) x (10) x (10)

  =  (√3) x (5) x (5)

  =  25 √3 cm²

Example 2 :

Find the length of the altitude of an equilateral triangle of side 3√3 cm.

Solution :

Side length of equilateral triangle (a)  =  3√3

Area of equilateral triangle  =  (√3/4) a²

  =  (√3/4) (3√3)²

=  (√3/4) (27)

Area of equilateral triangle  =  27√3 / 4   ---(1)

Here we should find the length of altitude,  so we use the formula base ⋅ height to find the area of equilateral triangle.

=   base ⋅ height

  =  (3√3/2) ⋅ h    ------(2)

(1)  =  (2)

(3√3/2) ⋅ h  =  27√3/4

h  =  (27√3/4) ⋅ (2/3√3)

h  =  9/2 

h  =  4.5

Hence the required height is 4.5 cm.

After having gone through the stuff given above, we hope that the students would have understood how we derive the formula for area of equilateral triangle and examples.

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